The energy distribution of subjets and the jet shape

Journal of High Energy Physics, Jul 2017

We present a framework that describes the energy distribution of subjets of radius r within a jet of radius R. We consider both an inclusive sample of subjets as well as subjets centered around a predetermined axis, from which the jet shape can be obtained. For r ≪ R we factorize the physics at angular scales r and R to resum the logarithms of r/R. For central subjets, we consider both the standard jet axis and the winner-take-all axis, which involve double and single logarithms of r/R, respectively. All relevant one-loop matching coefficients are given, and an inconsistency in some previous results for cone jets is resolved. Our results for the standard jet shape differ from previous calculations at next-to-leading logarithmic order, because we account for the recoil of the standard jet axis due to soft radiation. Numerical results are presented for an inclusive subjet sample for pp → jet + X at next-to-leading order plus leading logarithmic order.

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The energy distribution of subjets and the jet shape

Received: May The energy distribution of subjets and the jet shape Zhong-Bo Kang 1 2 4 5 6 7 8 9 10 Felix Ringer 1 2 3 5 7 8 9 10 Wouter J. Waalewijn 0 1 2 7 8 9 10 0 Nikhef, Theory Group 1 Los Alamos , NM 87545 , U.S.A 2 Los Angeles , CA 90095 , U.S.A 3 Nuclear Science Division, Lawrence Berkeley National Laboratory 4 Mani L. Bhaumik Institute for Theoretical Physics, University of California , USA 5 Theoretical Division, Los Alamos National Laboratory 6 Department of Physics and Astronomy, University of California , USA 7 Science Park 105 , 1098 XG, Amsterdam , The Netherlands 8 Science Park 904 , 1098 XH Amsterdam , The Netherlands 9 University of Amsterdam 10 Berkeley , CA 94720 , U.S.A We present a framework that describes the energy distribution of subjets of radius r within a jet of radius R. We consider both an inclusive sample of subjets as well as subjets centered around a predetermined axis, from which the jet shape can be obtained. R we factorize the physics at angular scales r and R to resum the logarithms of r=R. For central subjets, we consider both the standard jet axis and the winner-take-all axis, which involve double and single logarithms of r=R, respectively. All relevant one-loop matching coe cients are given, and an inconsistency in some previous results for cone jets is resolved. Our results for the standard jet shape di er from previous calculations at next-to-leading logarithmic order, because we account for the recoil of the standard jet axis due to soft radiation. Numerical results are presented for an inclusive subjet sample for pp ! jet + X at next-to-leading order plus leading logarithmic order. - For r 1 Introduction 2 Inclusive cone jets revisited 3 Inclusive subjets De nition of subjet function NLO calculation Renormalization and resummation of ln R Matching for r Central subjets for the winner-take-all axis Central subjet function for r . R Matching for r R and resummation of ln(r=R) Central subjets for the standard jet axis 6 The jet shape 7 Conclusions 1 Introduction Central subjet function for r . R Factorization for r R Resummation of ln(r=R) Inclusive subjets Winner-take-all axis Standard jet axis Relation with TMD fragmentation 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 4.1 4.2 5.1 5.2 5.3 6.1 6.2 6.3 6.4 In this paper, we study the energy distribution of subjets with radius r inside a jet of radius R, as illustrated in gure 1. We will consider jets de ned through the anti-kT algorithm [1] or an infrared and collinear-safe cone algorithm. Subjets are obtained by reclustering the particles inside the reconstructed jet with radius parameter r < R. In addition, we consider the subjet of radius r centered around the standard jet axis or the winner-take-all axis (WTA) [2]. We mostly focus on an inclusive jet sample pp ! jet + X, but brie y discuss how our framework can be extended when a veto on additional jets is imposed. { 1 { on describing the energy fraction zr of the jet that is carried by the subjet. HJEP07(21)64 Speci cally, we will develop the theoretical framework to calculate F (zr; r; ; pT ; R) = d where zr is fraction of the jet energy contained in the subjet of radius r, and and pT are the rapidity and transverse momentum of the jet with radius R. First proposed in ref. [3], the subjet distribution simultaneously provides information about the longitudinal and transverse energy distribution inside jets, through zr and r. The subjet observables considered in this work are connected to both the standard jet shape [4{7] and the jet fragmentation function [8{11]. On the one hand, the jet shape is the average value of zr as function of r, for subjets centered on the jet axis. On the other hand, the jet fragmentation function describes the longitudinal momentum (or energy) fraction of hadrons in the jet. This is the r ! 0 limit of the inclusive subjet energy fraction zr, where the collinear singularity is now cut o by hadronization instead of r. The jet shape [12{15] and the jet fragmentation function [16{18] have been measured by the LHC experimental collaborations in both proton-proton and heavy-ion collisions. We expect that similar measurements are feasible for the observables discussed in this work. There are several ways in which subjet distributions are valuable to present day collider phenomenology: rstly, the various distributions of subjets discussed in this work provide a powerful test of our understanding of perturbative QCD at very high energies. Studying the energy distribution of subjets probes both the longitudinal and the transverse momentum distribution within jets at a more di erential level, and may extend our current understanding of the underlying QCD dynamics. Secondly, the distribution of subjets can be used for discriminating QCD jets from boosted heavy objects, such as W bosons or top quarks. Their hadronic decays would produce two or three jets, which become the subjets of one fat jet due to their boost. This plays an important role in many searches for Beyond the Standard Model (BSM) physics [19{22]. Many of the taggers used for identifying such a two or three prong decay are quite sensitive to soft radiation [23{28]. For several of the subjet observables we consider, this soft sensitivity is power suppressed and collinear factorization is su cient. In addition to being theoretically more robust, a reduced sensitivity to soft radiation is also advantageous experimentally due to the messy LHC environment. Our work on the inclusive subjet distribution and the distribution of { 2 { subjets centered about a speci ed axis provides a rst step in the direction of taggers that are less sensitive to soft radiation. An example of a more direct connection to BSM searches is given in ref. [29], where the authors proposed to use jet shapes (\jet energy pro les") to search for new physics at the LHC. Another application in the context of jet substructure is the discrimination of quark and gluon jets using e.g. fractal observables de ned on subjets rather than hadrons [30]. Thirdly, the subjet distribution is particularly suited for measuring the modi cation of jets in heavy-ion collisions, see e.g. [31{33]. Jets that traverse the quark-gluon plasma get modi ed both in the longitudinal and transverse momentum direction, as can be collectively seen from the modi cation of longitudinal jet fragmentation function [17] and transverse jet energy pro le [14]. By identifying subjets inside a reconstructed jet, the correlations between these e ects can be studied in a single measurement. An advantage of using subjets over hadrons is that the subjet distribution does not require the additional non-perturbative input of fragmentation functions. Our setup relies on collinear factorization: rst we exploit that the radius R is small, to factorize the dynamics of the jet from the rest of the cross section.1 For pp collisions, we have d Here fa;b denote the parton distribution functions and Hacb are hard functions describing the production of an energetic parton of avor c with transverse momentum pT =z and rapidity with respect to the beam axis. The subjet functions Gcjet describe the subsequent conversion of that parton into a jet moving in (roughly) the same direction but with transverse momentum z pT =z = pT , containing a subjet of radius r with fraction zr of the jet energy. The argument !R = 2pT = cosh of Gcjet is the large light-cone component of the jet momentum, and the arguments r and R are suppressed. The symbols denote convolution products associated with the variables xa;b and z, which are explicitly written out in eq. (3.41). Power corrections to the factorized cross sections are order R2 suppressed. We will consider both r . R and r R. In the rst case, only single logarithms of the form sn lnn R need to be resummed to all orders. The subjet functions Gcjet follow timelike DGLAP evolution equations allowing for the resummation of logarithms in the jet size parameter R [3, 36, 37] (see refs. [38, 39] for a generating functional approach to jet radius resummation). For all subjet observables considered in this work, the resummation of the logarithms of R is the same. For r R, we encounter additional large logarithms of r=R. The structure and resummation for this class of logarithms depends on how the subjet of size r is identi ed. For an inclusive subjet sample, we perform an additional collinear factorization for the subjet, matching the subjet functions Gc function [3, 36] for the subjet. This enables us to resum single logarithms sn lnn(r=R) using jet onto a semi-inclusive jet another DGLAP type evolution equation. The refactorization for central subjets (i.e. those centered around a speci c axis) in the limit r R di ers from the inclusive subjet case, and crucially depends on the choice 1In practice this still works for rather large values of R. E.g. in refs. [34, 35] the error from the small R approximation remains below 5% for R = 0:7. { 3 { of axis. The standard jet axis is sensitive to soft radiation inside the jet, since the jet axis is aligned with the total jet momentum. By contrast, the winner-take-all axis is insensitive to soft radiation, but the location of the axis depends on the details of the collinear radiation. For the winner-take-all axis our factorization enables resummation to all-orders in perturbation theory using a (modi ed) DGLAP evolution [40], whereas for the standard jet axis this is complicated due to non-global logarithms. We will also calculate the average zr value from eq. (1.2) for the central subjet, which corresponds to the jet shape. Its cross section has a single logarithmic dependence on r=R for the winner-take-all axis and a double logarithmic dependence for the standard jet axis. Our factorization formula for the standard jet shape for r R di ers from earlier approaches [5{7].2 Speci cally, it involves a further refactorization to account for the soft radiation that recoils against the jet axis. The additional logarithms of r=R that we can resum, enter the cross section at next-to-leading logarithmic order. This is similar to the broadening event shape where the recoil of soft emissions on the direction of collinear particles (which only enters at NLL order) was initially overlooked [41] and only realized later [42]. Like in ref. [43], the e ect of recoil can be removed by using the winner-take-all axis. However, in this case this removes all soft sensitivity. The outline of our paper is as follows: in section 2 we revisit the calculation of the semiinclusive jet function, addressing an inconsistency in the literature for cone algorithms, and presenting corrected analytical results. We discuss the inclusive production of subjets in section 3 in terms of the subjet function, for both cone and anti-kT algorithms, and show numerical results for eq. (1.1) for pp ! (jet jr) + X at NLO+LLR+LLr=R. In sections 4 and 5 we focus on subjets centered on the winner-take-all axis and the standard jet axis, respectively. In all sections, r . R as well as r R are considered, and all matching coe cients are calculated at NLO. The jet shape is the second moment (average zr) of the result in sections 4 and 5, and can be directly related to TMD fragmentation, as discussed in section 6. We conclude in section 7 and provide an outlook. 2 Inclusive cone jets revisited In this section we review the calculation of the semi-inclusive jet functions (siJFs), which enter in the cross section for single inclusive jet production, pp ! jet + X. Speci cally, the cross section for inclusive jet production satis es the factorization theorem in eq. (1.2), after replacing Gc jet by the siJF Jc [36]. We rst address an inconsistency in the literature for cone algorithms, before considering the calculation of subjet functions in the following sections (as we also present results for cone algorithms there). Our default notation will be for e+e algorithms, where a jet is de ned in terms of its energy E = !R=2 and angle R. These results equally apply to pp algorithms, with the replacement !RR ! 2pT R in terms of a jet radius de ned in ( ; ) coordinates, see e.g. ref. [35]. For single inclusive jet production in proton-proton collisions at NLO, pp ! jet + X, there are either one or two nal-state partons inside the observed jet, whose possible 2From reading ref. [7] one may get the impression that they use a recoil-free axis. The authors con rm that this is not the case. { 4 { (A) the quark and gluon are inside the jet, (B) only the quark is inside the jet, (C) only the gluon is inside the jet. assignments are illustrated in gure 2. In refs. [11, 34, 36, 44{46] the cone algorithm was (e ectively) implemented for two nal-state partons as HJEP07(21)64 1 = 2q? ; x! 2q? (1 x)! ; { 5 { Partons in single jet: Partons in separate jets: 1 < R and 2 < R ; = 1 + 2 > R ; where 1 and 2 are the angles of the nal state partons with respect to the initiating parton. However, these regions of phase space are not complementary, and there are con gurations with R < < 2R that are double counted. The resolution depends on the speci cs of the cone algorithm. For example, if only the particles themselves are used as seeds for the cone algorithm, the rst criterion requires modi cation and the resulting algorithm happens to coincide with anti-kT (for two parton con gurations). For the midpoint [47] and the SISCone [48] algorithms, the correct implementation is Partons in single jet: Partons in separate jets: 1 < R and 2 < R ; 1 > R or 2 > R : Of course the midpoint and the SISCone algorithms will di er with additional particles. We now consider the calculation of the semi-inclusive quark jet function. The requirement that the partons are in separate jets in eq. (2.2), leads to the following expression for the case where the quark is inside the observed cone jet in gure 2(B), Z d 2 2c;q h (x < 1=2) ( 1 > R) + (x > 1=2) ( 2 > R)i (x z) : The corresponding expression when the gluon makes the observed jet, gure 2(C), is obtained by substituting (x matrix element in eq. (2.3) are given by z) ! (1 x z). The collinear phase space and (squared) Z d 2 2c;q = s (e E 2 ) Z 1 (1 ) 0 dx CF 1 + x2 1 x (1 x) Z qd1q+?2 ; ? where x is the momentum fraction and q? is the transverse momentum of (one of) the nal partons with respect to the initiating quark. The angles 1 and 2 can be expressed in terms of x and q? as follows (2.1) (2.2) (2.3) (2.4) (2.5) for the partons with momentum fraction x and (1 x). The angle between the partons is 2q? x(1 x)! : After evaluating the integrals in eq. (2.3) and combining it with the result when both partons are in the jet [11, 34, 36, 49], we obtain the new results for the cone semi-inclusive jet function: 2 s J cone(z; !R; ) = (1 q z) + LR Pqq(z) + Pgq(z) 2CF (1 + z2) The result for gluon-initiated jets can be obtained in a similar way, J cone(z; !R; ) = (1 z)+ g 2 s LR Pgg(z)+2nf Pqg(z) 4CA (1 z+z2)2 z J cone(z; !R; ) = Jqcone ref. [36](z; !R; ) q J cone(z; !R; ) = Jgcone ref. [36](z; !R; ) g + + 2 2 s s 2 Pqq(z)+Pgq(z) z > As the results for the semi-inclusive jet functions in ref. [36] are consistent with earlier analytical results for single inclusive jet production for cone algorithms in refs. [11, 34, 44, 45], the above equations also provide a correction to these earlier cone jet results. As a consistency check, we note that only the updated cone jet results in eq. (2.10) satisfy the following momentum sum rule introduced in [3]3 Z 1 0 dz z Ji(z; z!; ) = 1 ; (2.11) 3Ref. [3] only considered anti-kT jets, and thus did not verify the momentum sum rule for cone jets. { 6 { where the large momentum component of the initiating parton ! = !R=z is held xed. Similarly, the NLO matching coe cients for the jet fragmentation function as presented in [11, 46] for cone jets are modi ed as follows Jij cone;(1)(z; zh; !R; ) = Jij cone ref. [46];(1)(z; zh; !R; ) (2.12) zh) 2 s + (1 2Pji(z) z > z < For more details, we refer the interested reader to the earlier publications listed above. 3 Inclusive subjets In this section, we study the (semi-inclusive) subjet function (SJF), which describes the energy distribution of all subjets inside a jet as in eq. (1.2). We gives its de nition in section 3.1, and calculate it to next-to-leading order (NLO) in section 3.2. In section 3.3 we derive the renormalization group equation (RGE) of the subjet function, which we use to resum the logarithms of the jet radius R. We subsequently consider r R in section 3.4, performing the matching onto semi-inclusive jet functions (siJF) that describe the subjets of radius r, and use this to resum the large logarithms of r=R. The limit r ! R is discussed in section 3.5 and the fragmentation limit r ! 0 is considered in section 3.6. In section 3.7 we discuss the subjet function for exclusive jet production. We will drop the adjective \semi-inclusive" in front of the SJF, since we restrict ourselves to inclusive jet samples everywhere else. In section 3.8 we show numerical results for the momentum fraction of subjets in pp ! (jet jr) + X at NLO+LLR+LLr=R. 3.1 De nition of subjet function We de ne the subjet function as a matrix element in Soft-Collinear E ective Theory (SCET) [50{53]. In our de nitions and calculations e+e jet algorithms will be our default, which de ne a jet in terms of its energy E = !R=2 and angle R, and similarly for the subjet. Our results directly apply to pp algorithms with the replacement ER ! pT R, where the jet radius parameter R now refers to a distance in ( ; ) coordinates. The subjet functions for quark and gluon-initiated jets Gq jet and Ggjet are de ned as ;a(0)jXi zr !r !R ; (3.1) suppressing the dependence on r and R in the arguments. Here n = (1; n^) is a light-cone vector with its spatial component n^ along the jet axis, while n = (1; n^) is a conjugate { 7 { HJEP07(21)64 light-cone vector such that n2 = n2 = 0 and n n = 2. collinear quark and gluon elds in SCET. The sum over states jXi runs over all nal-state particles and includes their phase-space integrals. We sum over all reconstructed jets with radius parameter R in X and all subjets jr with radius r. The large light-cone momentum (approximately twice the energy) of the initiating parton, jet and subjet are denoted by !, !R and !r, respectively. For the eld producing the initiating parton this is encoded using the (label) momentum operator P. The variables z and zr describe the momentum fraction of the initiating parton carried by the jet, and that of the jet carried by the subjet, n and Bn? are gauge invariant and are thus given by 3.2 NLO calculation z = !R ! ; zr = !r !R We now calculate the subjet function for quark-initiated jets, Gqjet(z; zr; !R; ). For de niteness, we discuss the case where the anti-kT algorithm is used for reconstructing both the larger jet of size R and the subjet of size r, i.e. we consider \anti-kT -in-anti-kT ". However, at the end of this section we also present results for \cone-in-cone" and \cone-in-anti-kT ", see eq. (3.16). For \anti-kT -in-anti-kT " and \cone-in-cone" our calculations reveal that, at least at next-to-leading order, these results can be fully expressed in terms of known quantities by eq. (3.24). The calculation of the gluon subjet function Gg steps. Note that the jet algorithms anti-kT , kT [54, 55] and Cambridge/Aachen [56, 57] jet follows the same yield the same results to the order that we are considering. At leading order, the subjet functions are simply given by Gi jet;(0)(z; zr; !R; ) = (1 z) (1 zr) ; since the total energy of the initiating parton is transferred to the jet (of size R) and subjet (of size r). We perform the next-to-leading order calculation in pure dimensional regularization, where all virtual diagrams vanish. The collinear matrix element and phasespace were given in eq. (2.4). The ve possible assignments of the partons over the (sub)jet are shown in gure 3, which we discuss in turn: (A) The quark and gluon are inside the jet and subjet All the initial quark energy is transferred to the jet and subjet, so z = !R=! = 1 and zr = !r=!R = 1. This leads to (A) = (1 z) (1 zr) Z d 2 2c;q (r > ) ; where the -function encodes the constraint that both partons are inside the jet and subjet when using the anti-kT algorithm. Performing the q ? and x integrals and expanding in powers of , we nd (A) = (1 z) (1 zr) sCF 2 where Lr is de ned as We choose to write the results for all con gurations (A) - (E) in terms of !R. 1 2 + 3 2 + Lr + L2 3 2 { 8 { (3.3) (3.4) (3.5) (3.6) (A) (B) (C) (D) (E) and gluon are inside the jet and subjet, (B) only the quark is inside the subjet but both partons are in the jet, (C) only the gluon is inside the subjet but both partons are in the jet, (D) only the quark is inside the jet and subjet, (E) only the gluon is inside the jet and subjet. (B) Only the quark is inside the subjet but both partons are inside the jet The energy of the quark initiating the jet is transferred entirely to the jet z = 1, but only the fraction zr < 1 is contained inside the subjet. This contribution is The -function encodes the constraint that the partons are su ciently close together to be clustered into the jet but not so near that they are in the same subjet. Here LR is de ned in eq. (2.8) and Lr=R is given by Lr=R = Lr LR : Z 2 (B) = (1 z) d 2 2c;q (x zr) (R > > r) " = (1 z) s CF (1 zr) Lr=R LRLr=R Lr2=R 2 + 1+zr2 (1 zr) + Lr=R +O( ) : (C) Only the gluon is inside the subjet but both partons are inside the jet This con guration is analogous to (B) but exchanging the quark and gluon as shown in gure 3(C). This amounts to replacing (x zr) ! (1 x zr) in eq. (3.7), so where the quark splitting functions we use are de ned as Pqq(z) = CF 1 z + Pgq(z) = CF (D) Only the quark is inside the jet and subjet For the con guration shown in gure 3(D), only a fraction z < 1 of the initiating quark's energy is transferred to the jet of size R. However, all the energy of the jet 2 ; { 9 { (3.7) # (3.8) (3.9) (3.10) is inside the subjet, zr = 1. Thus its contribution to the SJF is given by (D) = (1 zr) d 2 2c;q (x z) ( > R) The only constraint from the jet algorithm is that both partons are far enough apart that they are not clustered together into the jet. (E) Only the gluon is inside the jet and subjet This is analogous to (D) but with the replacement (x z) ! (1 x z) in eq. (3.11), (E) = (1 zr) 2 s 1 + LR Pgq(z) 2 ln(1 z)Pgq(z) CF z + O( ) : (3.12) Summing up the leading-order result as well as the ve contributions at O( s), we obtain jet Gq;bare(z; zr; !R; ) = Gqjet;(0)(z; zr; !R; )+(A)+(B)+(C)+(D)+(E) All 1= 2 poles cancel in the sum as well as all double logarithms L2R and Lr2=R. The result at NLO always involves (1 z) and/or (1 zr), since the nal state consists at most of two partons, but this structure does not generalize to higher orders. The remaining 1= pole is a UV divergence that will be removed by renormalization and the resulting time-like DGLAP equation can be used to resum logarithms of R, as discussed in the next section. The result for the gluon SJF is: jet Gg;bare(z; zr; !R; ) = (1 z) (1 zr) + (1 zr) + LR Pgg(z) + 2nf Pqg(z) + (1 z)Lr=R [Pgg(zr) + 2nf Pqg(zr)] + (1 zr) (1 z) which involves the splitting functions Pgg(z) = 2CA (1 z) + + Pqg(z) = TF z2 +(1 z)2 : z +z(1 z) + 0 (1 z) ; CA 4CA z) 2 ; Gq cone-in-cone(z; zr; !R; ) cone-in-cone(z; zr; !R; ) = (1 z) (1 zr)+ s 2 2 2 r R Gq cone-in-anti-kT (z; zr; !R; ) (1 zr)LR[Pqq(z)+Pgq(z)] +2Pgq(z) ln(1 z)+CF + (1 z)[Pqq(zr)+Pgq(zr)] + + (1 z) (1 zr)CF r < 2Lr=R ln 1 These results agree with the general setup of ref. [3]. However, there the contributions from (A), (B), (C) are treated separately (factorized) from (D) and (E) (see e.g. their eq. (44)). This allows them to relate their expression to exclusive results. As these contributions are at the same scale, the factorization probably fails at higher order in s . For the semi-inclusive fragmenting jet function [46], a similar structure was obtained as in eqs. (3.13) and (3.14). However, this case involved an additional IR pole that cancelled in the matching onto the standard fragmentation functions. For the SJFs, this IR pole is regulated by the size of the subjet r leaving a single logarithmic dependence on the ratio r=R, i.e. Lr=R. In section 3.4, we are going to match the SJF onto a semi-inclusive jet function for the subjet. This will lead to another time-like DGLAP equation that can be used to resum logarithms of r=R. We conclude this section by giving the renormalized one-loop results for the cone-incone and cone-in-anti-kT SJFs 1 z + zr >max n r ; 1 o (Lr=R+2 ln zr) + r > ; R 2 r < CA R 2 + r > CA zr > max n r ; 1 o (Lr=R +2 ln zr) k k 2 s TF nf 18 r R Renormalization and resummation of ln R The renormalization and resulting RG equation of the subjet function is identical to that of the semi-inclusive jet function, because the additional measurement of the subjet does not modify the UV behavior. This also follows from consistency, since the SJFs and siJFs can be interchanged in factorization theorems. For completeness we still present the essential equations. The renormalization of the SJF is given by which leads to the following RG evolution equation Gi;bare(z; zr; !R; ) = X Z 1 dz0 jet z z0 z Zik z0 ; d Gijet(z; zr; !R; ) = X Z 1 dz0 d z z0 ik z0 z ; Gkjet(z0; zr; !R; ) ; Gkjet(z0; zr; !R; ) ; with the anomalous dimension matrix ij. From our NLO calculation, we immediately obtain ij(z; ) = s Pji(z) ; so the SJF satis es the usual time-like DGLAP evolution equations. The one-loop renormalized quark SJF is given by Gqjet(z; zr; !R; ) = (1 z) (1 zr) + (1 + (1 z)Lr=R [Pqq(zr) + Pqq(1 zr)] + CF (1 zr) (1 z) This result for G can be used when the jet radii of inner and outer jets are comparable, r . R. By evaluating it at the scale and evolving it with eq. (3.18) to the scale of the hard scattering R H !RR 2 !R 2 jet become large and require resummation as well. This is achieved by an additional factorization, which we discuss next. R we have the following matching equation to all orders in s Gijet(z; zr; !R; r; R; ) = j where Jj are the semi-inclusive jet functions describing the subjet of size r. In this equation we have explicitly shown the dependence on the jet radius R and the subjet radius r in the arguments of the functions, to highlight its structure. This matching equation is very similar to that for the matching of the semi-inclusive fragmenting jet function onto fragmentation functions. In fact, the matching coe cients Jij are the same, since they are independent of r and we can therefore safely take the fragmentation limit r ! 0. We have also veri ed this through a direct calculation, and therefore do not give these matching coe cients, but refer the reader to ref. [46] for anti-kT and section 2 for cone algorithms. Interestingly, our calculations reveal that there are no O(r2=R2) power corrections in eq. (3.23) at NLO. In fact, for anti-kT -in-anti-kT and cone-in-cone the NLO subjet function is fully determined by Gi jet(1)(z; zr; !R; r; R; ) = Ji(j1)(z; zr; !R; R; ) + (1 z) Jj(1)(zr; !r; r; ) : (3.24) For cone-in-anti-kT , this equation does receive corrections when r > R=2. Having performed the factorization in eq. (3.23), we can now resum the additional logarithms of r=R with the help of another DGLAP RG equation. The scale R in eq. (3.21) sets the large logarithms LR to zero in the matching coe cients Jij , whereas r !rr 2 using the RG equation to evolve the siJF from r to R, we resum the single logarithms of r=R. 3.5 We will now start from the regime r R, for which the logarithms of r=R are resummed, and consider the limit when the inner jet radius becomes large. It is fairly straightforward to do this, because the absence of power corrections in eq. (3.23) at this order. The aim is to gain an analytical understanding of the r ! R limit, for which we consider anti-kT -inanti-kT at leading-logarithmic accuracy. We start by observing that for zr < 1, the only terms in eq. (3.13) that contribute are since all other terms are proportional to (1 zr). Note that the role of the variables z and zr are fundamentally di erent, in the sense that z is an integration variable for the convolution with a hard function and zr is the measured external variable. As can be seen from eq. (3.26), the subjet cross section at NLO is directly proportional ln(r=R) and, therefore, it can be considered as a direct probe of the ln(r=R) resummation. In the limit r R, we refactorize the subjet function Gi jet in terms of matching coe cients and evolved siJFs, see eq. (3.23). In order to recover the NLO result in eq. (3.26) from the resummed result in the limit r ! R, we nd that it is su cient to consider the leading-order matching coe cients Gqjet(z; zr < 1; !R; ) = 2 s (1 z)Lr=R [Pqq(zr) + Pgq(zr)] ; (3.26) HJEP07(21)64 as well as the leading-order initial condition for the siJFs for the subjets Ji(j0)(z; zr; !R; R) = ij (1 z) (1 zr) ; Ji(0)(zr; !r; r) = (1 zr) : terms in Gijet. Including O( s) corrections in the matching or initial condition would generate O( s2) Using the techniques of refs. [36, 46], we solve the DGLAP equations associated with the resummation of both logarithms ln(r=R) and ln R in Mellin moment space. In order to perform the resummation, we take double Mellin moments of the subjet functions Gi Z 1 0 Z 1 0 Gijet(M; N; !R; ) = dz zM 1 dzr zr N 1 Gijet(z; zr; !R; ) : We only discuss the resummation of logarithms ln(r=R) associated with the variable zr and the Mellin variable N . (The DGLAP equation for resumming ln R was given in eq. (3.18) and is associated with the variable z and Mellin variable M .) The convolution of matching coe cients and the siJFs in eq. (3.23) turns into simple products X j Gijet(M; N; !R; ) = Jij (M; N; !R; ) Jj (N; !r; ) : The delta functions of the leading-order matching coe cients in eq. (3.27) integrate to 1 when taking moments, so the subjet function in Mellin space is given by the siJF evolved from r to R. For the quark siJF at LL accuracy, we nd Jq(N; !r; R) = 20 [Pqq(N)+Pgq(N)] ; (3.27) (3.28) jet (3.29) (3.30) (3.31) s( r) 2 0 ln R r : (3.33) where Pji(N ) are the leading-order Altarelli-Parisi splitting functions in Mellin N moment space and 0 is the rst coe cient of the QCD beta function. Inserting the leading-order solution for the running strong coupling constant 1 s( R) = 1 s( r) + 2 this becomes In the limit r ! R, 2 0 Jq(N; !r; r; R) = exp (Pqq(N ) + Pgq(N )) ln 1 + Jq(N; !r; r; R) = 1 + 2 s( R) Lr=R[Pqq(N ) + Pgq(N )] + O( s2) : Finally, we need to perform the double Mellin inverse transformation of the whole subjet function which is given by contour integrals in the complex M and N planes G jet(z; zr; !R; ) = Z dM CM 2 i z M Z dN CN 2 i r z N Gjet(M; N; !R; r; R; ) : The rst term in eq. (3.34) does not contribute to the cross section, since it does not contain poles in N . The second term in eq. (3.34) directly gives the NLO contribution for zr < 1 shown in eq. (3.26) above. Therefore, we have recovered the xed-order cross section from the resummed result in the limit of r ! R. We also veri ed this numerically in section 3.8. Note that the inverse with respect to N in eq. (3.35) can be taken directly as it is associated with the observed external variable zr. However the resulting expression for z still needs to be convolved with the hard function. 3.6 The fragmentation limit r ! 0 For su ciently small r, the scale !rr=2 of the semi-inclusive jet function with radius r becomes nonperturbative. At that point we can no longer speak of subjets but are really probing individual hadrons, and the subjet function should be replaced by a fragmenting jet function (inclusive in hadron species). This limit is continuous, since the matching coefcients are the same whether we match onto subjets or hadrons, as discussed in section 3.4. We stress that our conclusions also apply to inclusive jet cross sections, as we will study the nonperturbative corrections to the semi-inclusive jet function. We start by factorizing the siJF into a perturbative and nonperturbative component, Ji(zr; !; ) = z0 i Z dz0 J pert(zr0; !; )JiNP zr ; !r : z0 r (3.36) In contrast to the rest of the paper, we work in terms of the large momentum component ! of the initiating parton i, instead of !r = z! of the (sub)jet. Jipert is the perturbative result (which was calculated at NLO in refs. [3, 36]) and JiNP captures the nonperturbative corrections. From boost invariance (or reparametrization invariance [58]) we infer that N=3 N=2 20. 1/r 1/r JqNP to the semi-inclusive quark jet function for extracted from parton-level and hadron-level Pythia data, using e+e collisions at a center-ofmass energy of ! = 500 GeV, with the e+e anti-kT algorithm. In the left panel we restrict to large values of r and show a t to c=r2 (solid lines). In the right panel, the asymptotic approach to the fragmentation limit is shown. The nonperturbative corrections for N = 2 vanish due to eq. (2.11). the arguments ! and r appear in the combination !r, which was exploited in writing the arguments of JiNP. Since Jipert has the same anomalous dimension as the full Ji, this implies that JiNP is independent of . Note that this crucially relies on including the nonperturbative corrections through a Mellin convolution in eq. (3.36), since the anomalous dimension is only diagonal in Mellin space. Although JiNP is a two-dimensional nonperturbative function, we know its limits: !r ! 1 : !r ! QCD : JiNP(zr; !r) ! JiNP(zr; !r) ! (1 h zr) ; X Dih(zr; = !r=2) : (3.37) The rst line is the perturbative limit, for which the nonperturative corrections (but not JiNP) vanish. From the continuity of the r ! 0 fragmentation limit of subjets, it follows that the semi-inclusive jet function turns into the fragmentation function (summed over hadron species), as shown on the second line. We have extracted JqNP from Pythia [59], using parton-level and hadron-level inclusive jet spectra for e+e collisions at a center-of-mass energy of 500 GeV, with the e+e anti-kT algorithm. This implies that ! = 500 GeV (whereas !r varies). Instead of considering the full zr dependence we take Mellin moments, such that JqNP is the ratio of hadron-level and parton-level cross sections. The result is shown in gure 4 as function of 1=r. In the plot 1 perturbative limit !r ! 1, JqNP(N; !r) ! 1, so to visualize the nonperturbative e ects we JqNP. Also, JqNP(N = 2; !r) = 1, due to the momentum sum rule in eq. (2.11) (which relies on using ! rather than !r). In the left panel we limit ourselves to !r=2 > 10 GeV, for which it is reasonable to carry out a series expansion in 2 QCD=(!r). We nd that the linear term vanishes and the quadratic term ( t shown as solid line in left panel) describes the points very well. There is a slight discrepancy for large values of r, but in this regime the O(r2) corrections to the factorization theorem may no longer be negligible. In the right panel of gure 4 we focus on the nonperturbative regime, displaying the asymptotic h n= 2 !r !R : i (3.38) HJEP07(21)64 behavior. For !r=2 5 GeV the nonperturbative corrections deviate from the quadratic t by about 10%, whereas for !r=2 . 1 GeV, the jets essentially consist of single hadrons. Up to this point we have focussed entirely on inclusive jet production. However, one can also consider exclusive jet production, where additional jets are vetoed. The collinear (energetic) radiation is then forced to be inside the jet. Adding this additional restriction to the de nition of the subjet function in section 3.1, Gqjet(zr; !R; ) =16 3 X Tr In this case there is no collinear radiation outside the jet, which is why we replaced X by JR. Consequently, there is no dependence on z, since z = !R=! = 1 always. In the NLO calculation, the contributions in gure 3(D) and (E) are absent. This does not modify the dependence on zr and r, but removes the z dependence and introduces double logarithms of R. These logarithms of R can again be resummed using the RGE of the subjet function. However, instead of the convolution structure seen in eq. (3.18), the RGE of the subjet function is now multiplicative, d d Gi;excl(zr; !R; ) = i;excl(!R; R; ) Gij;eetxcl(zr; !R; ) : jet (3.39) The anomalous dimension i;excl is the same as for the unmeasured jet functions of ref. [49], and given in eq. (6.26) therein. The appearance of double logarithms of R indicate a sensitivity to soft radiation and so the factorization in eq. (1.2) must be modi ed to include a soft function. The matching for r R in section 3.4 onto semi-inclusive jet functions still holds, jet Gi;excl(zr; !R; r; R; ) = X Z 1 dzr0 j zr z0 Jij;excl(zr0; !R; R; ) Jj z0 r r but the matching coe cients Jij;excl are not the same as in the inclusive case. Rather, they are the same as those of the fragmenting jet functions for exclusive jet samples, which were calculated in ref. [60] for cone algorithms and in refs. [61, 62] for anti-kT . 3.8 Phenomenology for pp ! (jet jr) + X We present numerical results for the momentum fraction of subjets measured on an inclusive jet sample pp ! jet + X. In analogy with the hadron-in-jet calculations in proton-proton collisions presented in refs. [11, 46], we adopt the notation pp ! (jet jr) + X, where jr denotes a subjet of size r inside the larger jet of size R. The factorization formula for the 108 p√ps→= (1j3etTjerV)X, ,|ηR|<=10..26, r = 0.2, anti-kT 105 101 10−1 anti-kT with jet radius R = 0:6 and subjet radius r = 0:2, for representative LHC kinematics s = 13 TeV, j j < 1:2. Shown are the NLO+LLR+LLr=R results for four di erent intervals of the jet transverse momentum [25; 50]; [50; 100]; [100; 200]; [200; 500] GeV. subjet distribution in proton-proton collisions is given by d pp!(jet jr)X dpT d dzr = X Z 1 dxa a;b;c xamin xa where the sum on a; b; c runs over all relevant partonic channels. The PDFs are denoted by fa;b and the hard functions are given by Hacb, which have been calculated to NLO in refs. [44, 63]. For all numerical results presented in this section, we use the CT14 NLO set of PDFs [64]. The variables s, pT and correspond to the center-of-mass (CM) energy, the jet transverse momentum and the jet rapidity respectively. The hard functions depend on the corresponding partonic variables s^ = xaxbs, p^T = pT =z and ^ = ln(xa=xb)=2. The lower integration bounds xamin, xbmin and zmin can be written in terms of these variables and are listed for example in refs. [11, 46]. The subjet function Gcjet in eq. (3.41) is evolved to the hard scale pT by solving the DGLAP evolution equations associated with the logarithms ln R and ln(r=R). Numerically, we solve the DGLAP equations in Mellin moment space using the techniques developed in refs. [36, 46], which in turn are based on the evolution packages of refs. [65, 66]. We jointly resum both single logarithms ln R and ln(r=R) with a combined accuracy of \NLO+LLR+LLr=R". The evolved subjet function Gc jet is divergent for z ! 1. We can nevertheless perform the integrals in eq. (3.41) by adopting the prescription of ref. [67], as discussed in detail in ref. [36]. Note that the factorized form of the cross section in 103 η)102 r, zF(101 100 10−1 NLO+LLR+LLr/R 0.01 p√ps→= (1j3etTjerV)X, ,|ηR|<=10.2.6, r = 0.2, anti-kT pT [50,100] GeV r = 0.1 r = 0.2 r = 0.3 , , T T p 103 η)102 r, zF(101 100 10−1 NLO+LLR+LLr/R 0.01 zr 0.1 1 zr 0.1 1 by multiples of 10. Right: subjet distribution measured on an inclusive jet sample, using the same gure 5 and the pT bin of [50; 100] GeV. The distribution is shown for di erent values of the subjet radius: r = 0:05 (green dotted), r = 0:1 (black dashed), r = 0:2 (blue dotdashed) and r = 0:3 (red solid). eq. (3.41) is a purely collinear factorization, i.e. there is no soft function. All numerical results presented here are normalized by the total inclusive jet cross section, see eq. (1.1), for which we resum single logarithms of the jet size parameter ln R at NLO+LLR accuracy. For our numerical results we choose representative LHC kinematics, taking a CM energy of p s = 13 TeV and a rapidity range of j j < 1:2. Both the jet and the subjets are identi ed using the anti-kT algorithm. We choose a jet radius of R = 0:6 for the outside jet. In gure 5 we plot the momentum fraction zr for a subjet radius parameter of r = 0:2 for di erent bins of the transverse momentum pT of the jet. We multiply the results for the di erent pT bins by multiples of 10 for better visibility. One immediately notices that the plotted curves look like the QCD Altarelli-Parisi splitting functions. This behavior of the cross section can be most easily understood by looking at the xed order results for the subjet function in eq. (3.20). Only the terms ln(r=R)Pji(zr) have a non-trivial functional dependence on zr, as all other terms are proportional to (1 zr) and do not contribute at xed order. The ln(r=R) resummation modi es the distribution slightly, so it is not exactly the splitting function. We would like to point out an important di erence of the results for the subjet distribution compared to the distribution of light charged hadrons inside jets, as presented in for example refs. [11, 46]. When measuring an identi ed hadron inside jets, the distribution falls continuously as zh increases. However, as can be seen from gure 5, the distribution of subjets starts to rise again for su ciently large zr. Whereas it becomes increasingly unlikely to nd a hadron that carries a large fraction of the complete jet, a subjet with radius r < R may still contain most of the energy of the larger outside jet as long as r is not too small. In order to better see the dependence on the jet pT , we plot on the left panel of gure 6, the same curves as in gure 5 but without multiplying them by multiples of 10. We observe only a relatively small dependence on the jet transverse momentum. Next, we study the dependence of the subjet distribution on the subjet radius parameter r. In the right panel of gure 6, we show the momentum fraction of the subjet for zr = 0.01 zr = 0.1 zr = 0.8 zr = 0.1 zr = 0.8 zr = 0.95 10−1 10−2 T T p , , 102 ,)η101 r(z 100 F 10−1 10−2 pp → (jetjr)X, R = 0.6, anti-kT √s = 13 TeV, |η| < 1.2, NLO+LLR+LLr/R gures 5 and 6, and the jet pT bin [25; 50] GeV (left) and [100; 200] GeV (right). Four representative values of the ratio zr are shown: 0.01 (green dotted), 0.1 (black dashed), 0.8 (blue dot-dashed) and 0.95 (red solid). HJEP07(21)64 four di erent values of r, ranging from 0.05 to 0.3. We choose the same kinematics as in gure 5 and restrict the jet transverse momentum pT to the bin [50; 100] GeV. We nd a relatively strong dependence on r which is also due to the ln(r=R) resummation e ects. To make this point more clear, we show in gure 7 the dependence of the cross section for xed values of zr as a function of r=R. Results are shown for four values of zr, and the two panels corresponds to di erent bins for the jet transverse momenta. One notices again the strong dependence on r which can span two orders of magnitude. For small zr the curves increase continuously as r decreases, since one nds more and more subjets. However, for su ciently large zr the curves atten out as r becomes small and can even turn over. This behavior is more pronounced for the smaller jet pT interval of [25; 50] GeV, and arises because it is not possible to capture a very large energy fraction zr of the jet within only a narrow subjet. 4 Central subjets for the winner-take-all axis In this section we focus on the energy distribution of the subjet centered about the winnertake-all (WTA) axis [2]. In section 4.1 we treat the case r . R, which parallels the discussion in section 3. We discuss the factorization for r R and the resummation of logarithms of r=R in section 4.2. 4.1 Central subjet function for r . R The di erence between the standard jet axis and WTA axis resides in the merging step of a clustering algorithm (and is thus not de ned for cone jets). Speci cally, it chooses the axis to be along the most energetic of the two particles (or pseudojets) that are being merged. For the con guration of at most two partons in the jet, the winner-take-all axis is along the most energetic one. This can simply be accounted for by an additional factor sCF (1 zr) (1 z) Lr=R [Pqq(zr) + Pqq(1 zr)] 13 2 1 ; 1=2) compared to the O( s) calculation in section 3.2. For example, for quark jets G~qjet(z; zr; !R; ) = (1 z) (1 zr) + s (1 where the tilde for the SJF indicates that we are restricting to the central subjet about the winner-take-all axis. At higher orders there will be more partons inside the jets, leading to more signi cant di erences between the calculation for the central subjet and an inclusive sample of subjets. The renormalization of the central subjet function is the same as that of the semiinclusive jet function. This is immediate at O( s) from the above, but holds at higher orders because the rest of the factorization theorem does not depend on whether the energy fractions of the central subjet is measured or not. The central subjet function therefore satis es the DGLAP evolution equation d d ~jet(z; zr; !R; ) = Gi j X Z 1 dz0 s z z0 z Pji z0 Gj ~jet(z0; zr; !R; ) : (4.2) By evaluating G~i at its natural scale R of the hard scattering, the logarithms of R= H !RR=2 and evolving it to the scale H !R=2 4.2 Matching for r R and resummation of ln(r=R) R, the central subjet function will contain large logarithms of r=R that require resummation. In direct analogy to eq. (3.23), this resummation is accomplished by the following matching equation to all orders in perturbation theory ~jet(z; zr; !R; r; R; ) = Gi j X Zzr1 dzz0r0 J~ij (z; zr0; !R; R; ) J~j z0 r r We rst describe this factorization formula and then explain why it holds for the winner The object J~j onto which we match is not the semi-inclusive jet function Jj , which since we have e ectively taken R ! 1. J~j is de ned as describes the distribution of energy fractions for all subjets produced by a parton. Rather, it only picks out the energy fraction of the subjet centered on the winner-take-all axis. It also di ers from the central subjet function G~ijet because all partons are clustered together, J~j (zr; !r; ) = 16 3 X X 1 2Nc Tr n= 2 h0j (!R n P) 2(P?) n(0)jXihXj n(0)j0i zr : (4.3) !r : !R (4.4) s there are at most two partons and the winner-take-all axis is along the most energetic one, so once again J~(1)(zr; !r; ) = j zr > Jj(1)(zr; !r; ) : This implies that the one-loop matching coe cients in eq. (4.3) are given by J~i(j1)(z; zr; !R; ) = Gi h ~jet;(1)(z; zr; !R; ) z) J~j(1)(zr; !r; )i 1 + O 1 2 (1 = = zr > zr > 2 1 1 h jet;(1)(z; zr; !R; ) Gi 2 Ji(j1)(z; zr; !R; ) ; for J~j . The evolution of J~j from the following modi ed DGLAP equation, and thus directly related to those for the inclusive case in eq. (3.23). In eq. (4.3) the G~ijet on the left-hand side contains physics at angular scales R and r, that are factorized into the objects J~ij and J~j on the right-hand side. The validity of this equation at next-to-leading order follows immediately from the above. However, to use it for resummation requires the factorization to hold to all orders in s. In particular, the axis nding must factorize between the scales r and R, i.e. the axis cannot be sensitive to radiation at the jet boundary. This was shown in ref. [40] for the winner-take-all axis when using Cambridge/Aachen or anti-kT , in the context of transverse-momentum-dependent fragmentation. Having performed the factorization in eq. (4.3), we can now resum the additional logarithms of r=R with the help of another RG equation. The scale R !RR=2 is the natural scale for the matching coe cients Jij and the scale r = !rr=2 is the natural scale r to R sums the logarithms of r=R, and is described by r 2 R2 i (1 z) Jj(1)(zr; !r; ) (4.5) (4.6) (4.7) (4.8) d d J~jet(zr; !r; ) = i k X Z 1 dzr0 ~ik z0 zr z zr0 r ; J~jet(zr0; !r; ) : k From the NLO expressions in section 4.2 it follows that the one-loop anomalous dimensions ~ij are given by ~i(j1)(zr; ) = zr > s Pji(zr) : 1 2 5 Central subjets for the standard jet axis In this section, we discuss the energy distribution of the subjet of radius r centered about the standard jet axis. We start with r . R in section 5.1, which involves a similar calculation as in section 3.2. In section 5.2, we discuss the factorization for r R, which takes on a completely di erent form than in sections 3.4 and 4.2. In particular, the standard jet axis introduces a sensitivity to (the recoil of) soft radiation. We discuss how the double logarithms of r=R can be resummed in section 5.3. This factorization su ers from non-global logarithms, obstructing an all-orders resummation. 5.1 We start by introducing the function describing subjets centered about the standard jet axis. To distinguish it from the subjet functions of sections 3 and 4 we denote it by G^ijet. We remind the reader that our default notation is for e+e algorithms, and that the central subjet thus corresponds to a cone of opening angle 2r. On switching to pp algorithms this of course becomes a \cone" in ( ; ) coordinates. It is de ned by G^qjet(z; zr; !R; ) = 16 3 X h n= 2 Tr !R zr (5.1) for quark jets, and analogously for gluon jets. There is no sum over subjets jr in the jet, because we now restrict ourselves to the momentum fraction of the central subjet. The NLO calculation has the same ingredients as in section 3.2, but the phase-space restrictions for con guration (A) through (C) are modi ed because we restrict to the subjet centered on the jet axis, (A) = Z (B) = (1 zr) ( < R) ( 1 < r) ( 2 > r) ; x zr) ( < R) ( 1 > r) ( 2 < r) : (5.2) The angles 1 ; 2 and were given in eqs. (2.5) and (2.6). The contributions (D) and (E) are not modi ed. There is also a new (and irrelevant) contribution from the con guration where neither parton is inside the central subjet. Performing the calculation, and carrying out the renormalization in the MS scheme, we nd for the cone algorithm ^cone(z; zr; !R; ) Gq = (1 z) (1 zr)+ 2 s (1 zr)LR [Pqq(z)+Pgq(z)]+ (1 z) zr > Lr=R[Pqq(zr) +Pgq(zr)]+ (1 z) (1 zr)CF z > z < + <zr < [Pqq(zr)+Pgq(zr)] Lr=R +2 ln 7 2 2 3 1 2 1 2 1 zr zr ; 1 2 1 2 1 2 2 1 2 r R 1 2 R r+R r R r R r R z > z < (1 z) <zr < [Pgg(zr)+2nf Pqg(zr)] Lr=R +2 ln ; (5.3) (1 zr)LR Pgg(z)+2nf Pqg(z) + (1 z) zr > Lr=R [Pgg(zr) (1 zr)LR [Pqq(z)+Pgq(z)] 1 z + 1 2 HJEP07(21)64 ^cone(z; zr; !R; ) Gg and for the anti-kT algorithm ^anti-kT (z; zr; !R; ) Gq z 1 z + + (r < R=2) (1 z) (1 zr) CA +4nf Pqg(z) ln(1 z)+TF z(1 z) 36 + r R 1 2 2 3 +2Pgq(z) ln(1 z)+CF + (1 z) zr > [Pqq(zr)+Pgq(zr)](Lr=R +2 ln zr) + (r < R=2) (1 z) (1 zr)CF (1 z) < zr < 1 [Pqq(zr)+Pgq(zr)] Lr=R +2 ln(1 zr) + (r > R=2) (1 z) (1 zr) +4CALi2 1 (1 z) [Pgg(z)+2nf Pqg(z)] Lr=R +2 ln zr (1 z) < zr < 1 [Pgg(z)+2nf Pqg(z)] Lr=R +2 ln(1 zr) + (r > R=2) (1 z) (1 zr) CF + 2 ^anti-kT (z; zr; !R; ) Gg (1 z) (1 zr)LR Pgg(z)+2nf Pqg(z) 3 2 Lr=R + 2 Lr=R 2Lr=R ln 1 r R r R ; 2 CA Lr2=R + 0 2 Lr=R 2CALr=R ln 1 +CA 8r R r 2 4r3 R2 + 9R3 +TF nf 3 r R R2 + 8r3 9R3 (5.4) Mode: hard(-collinear) collinear (collinear-)soft Scaling ( ; +; ?) !(1; R2; R) !(1; r2; r) !(r=R; rR; r) The renormalization of the central subjet function is again the same as that of the semiwhich enables the resummation of logarithms of R. R, the central subjet function contains large logarithms of r=R that require resummation. This is achieved through a second factorization, Z ^jet(z; zr; !R; r; R; ) = Hij (z; !RR; ) Gi d2k? Cj (zr; !rr; k?; ; ) Sj (k?; R; ; ) 1 + O r R ; R (5.6) where we made the dependence on r and R explicit in the arguments. We rst describe the factorization formula, which di ers signi cantly from eqs. (3.23) and (4.3), and then justify it. The hard function Hij describes how the energetic parton i produces a jet initiated by parton j with longitudinal momentum !R and jet radius R, carrying a momentum fraction z of parton i. The collinear function Cj describes the fraction zr of collinear radiation produced by parton j, within an angle r of the standard jet axis. It takes into account that the initial collinear parton has a transverse momentum k ? with respect to the jet axis, due to the recoil against the soft radiation, encoded in the soft function Sj . The transverse momentum dependence causes the factorization in eq. (5.6) to su er from rapidity divergences that require regularization. We will employ the -regulator [68, 69], for which denotes the corresponding rapidity renormalization scale. Other choices are possible too, see e.g. refs. [70{74]. The physical justi cation of eq. (5.6) is that the hard(-collinear) radiation cannot undergo a perturbative splitting inside the jet. Such a splitting would have a typical opening angle of order R and the contribution of such con gurations to the central subjet of radius r R is power suppressed. (Generically, neither of the partons would lie within the central subjets.) Perturbative splittings outside the jet are of course allowed and encoded by the z dependence of the hard function. Collinear splittings inside the jet that a ect the central subjet will have typical angle r, and are describe by the collinear function. { 25 { zr 1 2Nc !r ; !R Tr h n= 2 (5.7) The (collinear-)soft radiation is not energetic enough to in uence the zr measurement, but its transverse momentum k? a ects the jet axis, since the total transverse momentum with respect to the jet axis is zero, and must be taken into account. In the language of SCET, these correspond to distinct degrees of freedom with the parametric scaling of momenta summarized in table 1. The collinear function has the following de nition for j = q, Cq(zr; !rr; k?; ; ) = 16 3 X n P) 2(P? k?) n(0)jXihXj n(0)j0i i 1 Nc Xs and similarly for j = g. This describes the momentum fraction zr of the central subjet centered on the n axis. The recoil of the collinear radiation with respect to the jet axis due to soft radiation is taken into account through the 2 (P? The de nition of the soft function for j = q is given by Sq(k?; R; ; ) = Xh0jT[YnyYn] jXihXjT[YnyYn]j0i k ? ( i < R) ki;? : (5.8) The delta function sums the transverse momentum ki;? of soft radiation inside the jet, i < R. Yn is a soft Wilson line in the fundamental representation along the light-like direction n = (1; n^) of the jet, ig Z 1 0 Yn(x) = P exp dt n As(t n ) (5.9) and Yn is along the opposite direction n = (1; n^). For j = g the Wilson lines are in the adjoint representation and the overall normalization is modi ed 1=Nc ! 1=(Nc2 hard function Hij does not have a direct matrix element de nition in SCET, but instead 1). The it is de ned by the matching relation in eq. (5.6). The factorization for the standard jet axis in eq. (5.6) does not account for nonglobal logarithms (NGLs) [75, 76]. These arise because the transverse momentum of the (collinear-)soft radiation inside the jet is probed, but is unconstrained outside the jet.4 The tree-level hard, collinear and soft functions are given by X i2X i Hij (z; !RR; ) = ij (1 z) ; Cj (zr; !rr; k?; ; ) = (1 zr) jk?j < Sj (k?; R; ; ) = 2(k?) : !rr 2 ; (5.10) We calculate the one-loop corrections in pure dimensional regularization, such that the virtual corrections are scaleless and vanish. The contributions to Hij come from perturbative splittings of the parton i where the jet consists solely of parton j. For Hq(q1) and Hq(g1) 4Boosting to the frame where the jet becomes a hemisphere, the modes in table 1 become the standard SCETII hard, collinear and soft modes. Emissions into the other hemisphere are unconstrained and lead to additional (collinear-)soft Wilson lines [77, 78]. In the original frame these corresponds to emissions outside the jet described by Hij. these can directly be read o from diagrams (D) and (E) in section 3.2. For the anti-kT Hq(q1);anti-kT (z; !RR; ) = s CF (1 Hq(g1);anti-kT (z; !RR; ) = Hg(1q);anti-kT (z; !RR; ) = Hg(1g);anti-kT (z; !RR; ) = 2 2 2 2 LR + 2 12 ln (1 1 2 ln (1 2 ln (1 z)) Pgq (z) CF z] ; z)) Pqg (z) TF 2z (1 z) ; (1 + LRPgg (z) L2R CA 2 4CA 1 2 LR + CA 12 z + z2 2 0 2 z ln (1 1 z z) + (5.11) Similarly, the results for the cone algorithm can be written as Hi(j1);cone(z; !RR; ) = Hi(j1);anti-kT (z; !RR; ) + 2 s 2Pji(z) 1 2 1 2 z > z < ln(1 z) : (5.12) We next consider the soft function, which measures the transverse momentum of soft radiation in the jet. Performing the calculation using ref. [79], and noting that the jet region corresponds to rapidity y > ln(R=2) with respect to the jet axis, we obtain Sq(1) (k?; R; ; ) = 2 2 1 2 ln k2 = 2 ! ? k2 = 2 ? + + 1 1 2 k2 = 2 ? + ln 2R2 4 2 2 12 The result for Sg(1) follows by replacing CF ! CA. The full collinear function is already complicated at NLO, because it involves two measurements.5 As our current approach is anyway limited to NLL order due to nonglobal logarithms, we simply consider k ? = 0 (from the tree-level soft function). The calculation of the collinear function involves a slight modi cation to contributions (A), (B) and (C) to the central subjet function in section 5.1. Since r R, the collinear radiation is close to the center of the jet and does not probe the jet boundary, removing the ( < R) and (1 z) in eq. (5.2). For the quark case, the individual contributions are given by ~k 2 ? (5.13) (A) = Z d 2 2c;i (1 zr) ( 1 < r) ( 2 < r) 1 2 3 2 Lr + L2 3 2 5This seems similar to the case of jet broadening, for which the collinear contribution at one loop was only calculated in ref. [80]. HJEP07(21)64 2 Z (B) = d 2 2c;i (x zr) ( 1 < r) ( 2 > r) (1 zr)CF zr > 1 2 4CF +Lr ln(1 zr) 1 zr 1 2 + Lr + +2CF (1+zr) ln(1 zr)+2Pqq(zr) ln zr +O( ; ) ; 2 zr > 2 1 h2Pgq(zr) ln zr 1 zr +O( ) : i (5.14) Here we needed to include the -regulator for contribution (B). Adding up the various contributions and performing the renormalization, Cq(zr; !rr; 0; ; ) = (A) + (B) + (C) (1 2 ln + Lr + = 2 s 2 + s zr > 1 2 zr > 1 2 !R 2CF (1 + zr2) + 2 Pqq(zr) + Pgq(zr) ln zr A similar calculation yields the gluon collinear function Cg(zr; !rr; 0; ; ) = + 2 Pgg(zr) + Pqg(zr) ln zr 4CA !R (1 zr 2 : + 0 Lr + zr + zr2)2 ln(1 zr) 1 zr + 3 2 : ln(1 1 7 2 zr) zr 7 24 2 3 + 2 3 2Pgq(zr) ln(1 zr) 23 24 CA + +ln 2 0 2Pqg(zr) ln(1 zr) (5.15) (5.16) (5.17) We have veri ed that these one-loop ingredients indeed satisfy eq. (5.6). This check involved a subtlety related to distributions: since 1 2 = ; a naive expansion of G^icone in r=R leads to improperly regulated plus distributions such zr)+ instead of [ln(1 zr)]+. Rather, we verify eq. (5.6) by rst considering zr < 1 and then taking a suitable integral containing the point zr = 1, before expanding in r=R. Note that the di erence between G^icone and G^ianti-kT is completely captured by eq. (5.12) for r R. 5.3 The resummation is achieved by evaluating each of the ingredients in eq. (5.6) at their natural scale and evolving them to common scales and using their RG equations d d d d d d qHq(z; !RR; ) = qHg(z; !RR; ) = gHg(z; !RR; ) = gHq(z; !RR; ) = qC (!R; ; ) = gC (!R; ; ) = S q ( ; R) = S g ( ; R) = q (k?; ) = g (k?; ) = H C S ; k z X Z 1 dz0 H z z0 ik z0 3 2 1 ; s s Pqg(z); 2 ln 4 2 sCF ln 2R2 4 2 sCA ln 2R2 sCF 1 sCA 1 !R !R 1 1 ? ? + + ; ; 2 (k2 = 2) 2 (k2 = 2) + + ; : Hij (z; !RR; ) = Hkj (z0; !RR; ) : d d d d Ci(zr; !rr; k?; ; ) = iC (!R; ; ) Ci(zr; !rr; k?; ; ) : Si(k?; R; ; ) = iS( ; R) Si(k?; R; ; ) ; Ci(zr; !rr; k?; ; ) = i (k?; ) Ci(zr; !rr; k?; ; ) : Si(k?; R; ; ) = i (k?; ) Si(k?; R; ; ) : To avoid a cumbersome complication for resummation in momentum space [81], it is much more convenient to carry out the rapidity resummation in impact parameter space.6 The one-loop anomalous dimensions directly follow from the expressions in section 5.2, LR (1 z) + Pqq(z) ; CALR (1 z) + Pgg(z) ; (5.18) (5.19) HJEP07(21)64 (5.20) conservation. the following as they should yield the anomalous dimension of the central subjet function, see eq. (5.6). This is indeed the case, since the splitting function contributions in iHj remain uncancelled. 6 The jet shape In this section, we consider the jet shape which is the average momentum fraction of the central subjet distribution. For comparison, we also consider the inclusive subjet sample. The average momentum fraction for inclusive subjets simply amounts to a sum rule, in contrast to the case of central subjets. Speci cally, averaging the SJFs Gi back the semi-inclusive jet functions jet over zr, we get 0 dzr zr Gijet(z; zr; !R; ) = Ji(z; !R; ) : (6.1) This result holds both for all jet algorithms and for i = q; g, as it is simply due to momentum Applying this to the r R limit, described by the matching in eq. (3.23), we nd 0 0 dzr zr Gijet(z; zr; !R; ) = dzr zr Jij (z; zr; !R; ) dz0 z0Jj (z0; !r; ) : (6.2) This equation can be veri ed by using the momentum sum rule for the siJF in eq. (2.11) and combining it with the momentum sum rule for the fragmenting jet function [9, 46] 0 Z 1 0 dzr zr Jqq(z; zr; !R; ) + Jqg(z; zr; !R; ) = Jq(z; !R; ) ; dzr zr Jgg(z; zr; !R; ) + 2nf Jgq(z; zr; !R; ) = Jg(z; !R; ) : (6.3) In particular, by averaging over zr, all logarithms Lr=R in Gi the case for central subjets, as discussed in the following sections. jet disappear. This will not be 6.2 Winner-take-all axis The zr averaged results for the SJFs for central subjets along the WTA axis are given by: dzr zr G~qjet(z; zr; !R; ) = Jq(z; !R; ) + (1 dzr zr G~gjet(z; zr; !R; ) = Jg(z; !R; ) + (1 z) sCF Lr=R 2 2 s Lr=R CA 96 3 8 43 2 ln 2 ; 2 ln 2 (6.4) 7 TF nf 48 : In this case we have a single logarithmic dependence on Lr=R. The above expressions hold for both the cone and the anti-kT algorithm, as the algorithm-dependent pieces of G~ijet are contained in Ji. X Z 1 0 0 dzr zrG~ijet(z; zr; !R; ) = dzr zrJ~ij (z; zr; !R; ) dz0 z0J~j (z0; !r; ) ; (6.5) in direct analogy with eq. (6.2). However, the sum rules in eqs. (2.11) and (6.3) do not hold for J~j and J~ij due to the additional theta functions, which is why the single logarithms of For r R, we have 0 r=R persist. Standard jet axis 0 0 6r R 3r2 2R2 ; + CA + TF nf 203 36 41 18 8r R 4r R + 3 ln 1 + 2 3 R r R 2 0 ln 2 CA 3 R 3 ln 2 + 4Li2 + 2 ln2 1 + Here we present the zr averaged results for subjets along the standard jet axis, showing separate results for the cone and anti-kT algorithm. We obtain HJEP07(21)64 dzr zr G^qanti-kT (z; zr; !R; ) = Jqanti-kT (z; !R; ) + (1 z) sCF 2 1 2 2 Lr=R + 3 Note that here we have a double logarithms of Lr=R and that these expressions contain power corrections of the form r=R. We have compared this with results available in the literature: combining the in-jet calculation of ref. [7] (see also refs. [ 5, 6, 84 ] for earlier results obtained within standard QCD) with the out-of-jet contribution of ref. [36], we nd agreement with eq. (6.6). Also, for r = R these results reduce to the semi-inclusive jet function in ref. [36]. dzr zr G^ganti-kT (z; zr; !R; ) = Jganti-kT (z; !R; ) + (1 dzr zr G^qcone(z; zr; !R; ) = Jqcone(z; !R; ) + (1 dzr zr G^gcone(z; zr; !R; ) = Jgcone(z; !R; ) + (1 z) s 2 3r2 R2 + 8r3 9R3 3r2 R2 16r3 9R3 + z) sCF 2 3r r + R r + R z) s 2 CA Lr2=R + 0 2 Lr=R 2 4R4 2R4 2 Lr=R + 3 2 Lr=R + 3 2 ; R + 2CA ln2 1 + + 0 ln 1 + + 4CA Li2 + TF nf 3(r + R)3 2r2 + 4rR + 3R2 : (6.6) 2 R CA Lr2=R + 0 2 Lr=R r r + R R + CA 6(r + R)3 11r2 + 22rR + 12R2 0 Our refactorized cross section in the limit r R and, hence, the resummation of logarithms Lr=R for the jet shape takes on a di erent form than in the literature. From eq. (5.6) it follows that dzr zr G^ijet(z; zr; !R; r; R; ) = Hij (z; !R; R; ) dzr zr Cj (zr; !rr; k?; ; ) Sj (k?; R; ; ) 1 + O Z h h Z 1 R 0 ; R i (6.7) (6.9) where the refactorization of the soft (recoil) contribution is the new ingredient. This additional factorization is essential to resum the logarithms of r=R beyond LL accuracy. Relation with TMD fragmentation Because averaging is linear, the jet shape can directly be related to TMD fragmentation through the following sum rule Z dzr zr G~ijet(z; zr; !R; ) = X Z h jk?j !Rr=2 d2k dzh zh G~ih(z; !R; k?; zh; ) ; (6.8) and similarly for the standard jet axis (replacing tildes by hats). This formula describe the central subjet as the sum of the contributions of its hadron constituents. The TMD fragmentation function G~ih(z; !R; k?; zh; ) is the number density of hadrons of species h, momentum fraction zh and transverse momentum zhk? with respect to the winner-take-all axis [40]. The restriction to the central subjet of radius r is encoded by h zhjk?j = We have veri ed that eq. (6.8) holds for anti-kT with the winner-take-all axis as well as the standard jet axis using the one-loop results in refs. [40] and [85, 86]. 7 In this paper we considered the energy fraction zr of subjets of size r inside a jet of size R. We presented analytical results for the following three cases: inclusive subjets obtained by reclustering all particles in the jet with jet radius parameter r, as well as central subjets along the winner-take-all axis and along the standard jet axis. The single logarithms of the form sn lnn R are the same in each case and can be resummed to all orders by solving the associated DGLAP evolution equation. We also considered the logarithms of the ratio of the jet size parameters ln(r=R), whose structure depends on the particular subjet observable. For each case, we performed an additional refactorization of the cross section in the limit r R, enabling the resummation for this class of logarithms. For central subjets along the WTA axis, this refactorization is known to all-orders in s but we are currently restricted to leading logarithmic resummation because of our knowledge of the anomalous dimensions. For central subjets along the standard jet axis, an all-orders factorization formula is hindered by non-global logarithms. We presented numerical results for the zr-distribution of inclusive subjets measured on an inclusive jet sample pp ! (jet jr)X, leaving numerical results for central subjets to future work [87]. In addition, we considered the average energy fraction of these results, which is known as the jet shape for subjets centered on the standard jet axis. For the jet shape, our factorization formula in the limit r R involves an additional refactorization compared to the literature, to account for the recoil e ect of soft radiation on the jet axis. Along the way, we also pointed out an inconsistency in the literature for analytical results for inclusive cone jets and their substructure. There are various possible applications of our work in the future. First of all, it will be very interesting to perform numerical calculations for central subjets along the two axes we considered in this work. This will be particularly relevant since our factorization for the standard jet shape in the limit r R di ers from that in the available literature at next-to-leading logarithmic order, and it is possible to compare to experimental data in this case. In addition, for the more di erential case (the zr-dependent case), experimental measurements will be feasible which can shed new light on the substructure of jets at the LHC. In addition, it will be interesting to further explore the possibility of how these \relatively inclusive" jet substructure observables can be used to discriminate between QCD jets and boosted objects. Finally, we expect that the di erent energy distributions of subjets considered in this work can be very relevant to better understand the properties of the quark-gluon plasma created in heavy-ion collisions. Acknowledgments We thank Y.-T. Chien, H. n. Li, X. Liu, D. Neill, V. Rentala and G. Sterman for discussions. We also thank T. Kaufmann, I. Vitev, and W. Vogelsang for comments on the manuscript. This work is supported by the National Science Foundation under Contract No. PHY1720486, the U.S. Department of Energy under Contract Nos. DE-AC02- 05CH11231, DE-AC52-06NA25396, by the Laboratory Directed Research and Development Program of Lawrence Berkeley National Laboratory, by the ERC grant ERC-STG-2015- 677323 and the D-ITP consortium, a program of the Netherlands Organization for Scienti c Research (NWO) that is funded by the Dutch Ministry of Education, Culture and Science (OCW). Open Access. 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Zhong-Bo Kang, Felix Ringer, Wouter J. Waalewijn. The energy distribution of subjets and the jet shape, Journal of High Energy Physics, 2017, 64, DOI: 10.1007/JHEP07(2017)064